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Minimizing actuator-induced errors in active space telescope mirrors The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Smith, Matthew W., and David W. Miller. “Minimizing actuatorinduced errors in active space telescope mirrors.” Space Telescopes and Instrumentation 2010: Optical, Infrared, and Millimeter Wave. Ed. Jacobus M. Oschmann et al. San Diego, California, USA: SPIE, 2010. 773122-14. © 2010 SPIE As Published http://dx.doi.org/10.1117/12.855947 Publisher SPIE Version Final published version Accessed Wed May 25 18:21:23 EDT 2016 Citable Link http://hdl.handle.net/1721.1/61645 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms Minimizing actuator-induced errors in active space telescope mirrors Matthew W. Smith and David W. Miller Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139 ABSTRACT The trend in future space telescopes points toward increased primary mirror diameter, which improves resolution and sensitivity. However, given the constraints on mass and volume deliverable to orbit by current launch vehicles, creative design solutions are needed to enable increased mirror size while keeping mass and volume within acceptable limits. Lightweight, segmented, rib-stiﬀened, actively controlled primary mirrors have emerged as a potential solution. Embedded surface-parallel actuators can be used to change the mirror prescription onorbit, lowering mirror mass overall by enabling lighter substrate materials such as silicon carbide (SiC) and relaxing manufacturing constraints. However, the discrete nature of the actuators causes high spatial frequency residual errors when commanding low-order prescription changes. A parameterized ﬁnite element model is used to simulate actuator-induced residual error and investigate design solutions that mitigate this error source. Judicious speciﬁcation of mirror substrate geometry and actuator length is shown to reduce actuator-induced residual while keeping areal density constant. Speciﬁcally, a sinusoidally-varying rib shaping function is found to increase actuator inﬂuence functions and decrease residual. Likewise, longer actuators are found to oﬀer reduced residual. Other options for geometric shaping are discussed, such as rib-to-facesheet blending and the use of two dimensional patch actuators. Keywords: Active mirrors, actuator-induced residual error, ﬁnite element modeling, surface-parallel actuators, cryogenic SiC mirrors. 1. INTRODUCTION Space-based optical telescopes, like their ground-based counterparts, are typically designed with an emphasis on maximizing aperture diameter. Larger apertures allow increase telescope resolution and sensitivity, two critical ﬁgures of merit for high quality imaging. However unlike ground systems, space telescopes are severely constrained by launch vehicle mass and volume limits. This necessitates creative design solutions for increasing aperture area with minimal corresponding mass and volume growth. Current approaches are the use of a segmented primary mirror that launches in a compact stowed conﬁguration and unfolds on orbit—addressing the volume constraint—and low areal density mirror design—addressing the mass constraint. The challenge then becomes preserving optical performance in the presence of these design changes. Thus designing compact, lightweight, yet optically precise primary mirrors for space telescopes presents a signiﬁcant challenge. This work oﬀers solutions, particularly for the mass-constrained (i.e. low areal density) case. Over the past few decades, reaction bonded silicon carbide (SiC) has emerged as a promising material for space telescope mirrors.1 It has a relatively low coeﬃcient of thermal expansion (CTE), high thermal conductivity, a high elastic modulus, and relatively low density.2 When an SiC substrate is augmented by embedded surfaceparallel electrostrictive actuators, the result is a highly integrated active mirror system that can undergo changes in optical prescription—e.g. changes in radius of curvature (RoC)—without the need for a bulky reaction structure.3 These actuators allow for on-orbit correction of ﬁgure errors due to CTE eﬀects in a changing thermal environment, or due to RoC mismatch from the manufacturing process. Further author information: (Send correspondence to M. Smith) M. Smith: E-mail: m [email protected], Telephone: 1 626 665 3201 D. Miller: E-mail: [email protected], Telephone: 1 617 253 3288 Space Telescopes and Instrumentation 2010: Optical, Infrared, and Millimeter Wave, edited by Jacobus M. Oschmann Jr., Mark C. Clampin, Howard A. MacEwen, Proc. of SPIE Vol. 7731, 773122 · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.855947 Proc. of SPIE Vol. 7731 773122-1 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms 1.1 Actuator-induced residual error While highly integrated active SiC mirrors oﬀer tremendous promise in terms of low areal density and manufacturing cost, complications arise as a result of actuator behavior. Speciﬁcally, the discrete placement and local inﬂuence of the actuators makes it diﬃcult to command perfectly smooth low-order changes such as RoC adjustments. This is depicted in Figure 1. (a) (b) (c) (d) Figure 1. (a) Edge-on view of an un-deformed mirror with upward concavity due to its radius of curvature, (b) desired surface change for positive ΔRoC, (c) actual surface change for positive ΔRoC with discrete actuators, and (d) actuatorinduced residual error (diﬀerence of b and c). . Figure 1(a) shows a rib stiﬀened active mirror segment from an edge-on view. The reﬂecting surface is on the top of the mirror in this view, with slight upward concavity due to a parabolic optical ﬁgure. A phasing maneuver may require an increase in RoC, which is equivalent to seeking the mirror surface displacements shown in Figure 1(b): for an increase in RoC, the mirror is made “ﬂatter” by commanding the edges to deﬂect downward. Because of the discrete locations and localized inﬂuence of the actuators, however, the actual displacement is the shape shown in Figure 1(c). Instead of being smooth, the actual displacements more resemble the surface of a golf ball, with the actuators introducing an unwanted high spatial frequency variation on top of the desired low order shape. The diﬀerence between the optically perfect desired surface change and the actual surface change is termed actuator-induced residual error, or actuator “quilting”. This residual error acts as an optical aberration, degrading the image quality. 1.2 Objective Thus the challenge is to preserve the ability to command low-order shape changes while simultaneously reducing high spatial frequency residual error caused by the actuators. If actuator-induced residual is too large, the operational beneﬁts from having an adjustable mirror prescription are quickly negated. Furthermore, any residual mitigation should ideally be accomplished without adding to the mass or complexity of the mirror system. Recent work4, 5 points to creative design of mirror geometry via ﬁnite element modeling as one possible way to decrease actuator-induced residual. Therefore the objective of this work is the to reduce actuator-induced high spatial frequency residual error by manipulating mirror geometry using a parametric ﬁnite element model, while keeping areal density and number of actuators constant. Proc. of SPIE Vol. 7731 773122-2 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms 2. PRIOR WORK The eﬀort described here draws on prior research using the Modular Optical Space Telescope (MOST) ﬁnite element model developed at the MIT Space Systems Laboratory (SSL). A high ﬁdelity, parameterized representation of lightweight space telescope primary mirrors, MOST allows for the rapid generation and evaluation of rib-stiﬀened, actively controlled designs across the trade space. Cohan6, 7 uses the MOST model along with vibroacoustic analysis to determine optimal mirror designs for launch survival and on-orbit performance. Likewise, Gray4, 8 considers high-spatial frequency errors due to actuator and manufacturing eﬀects. After determining the parameters with the largest inﬂuence on overall residual error, optimizations are carried out to derive a set of optimal design relationships that minimize uncorrectable high spatial frequency error while satisfying manufacturing constraints. Much of the modeling infrastructure used in the present work has heritage from the above MOST investigations. Finite element (FE) modeling has been previously employed by a number of authors to analyze mirror design. Nied and Rudmann9 use FE models to characterize thermal distortions in the Hubble Space Telescope (HST) primary mirror. Shepherd et al.10 construct an FE representation of a actively controlled membrane mirror, and use the model to validate a novel modal transformation method for describing mirror displacements. Finally, Jordan and Miller11 use an FE model to demonstrate the feasibility of mirror ﬁgure control using embedded strain gauges and temperature sensors. These simulation-based results complement a large body of prior work on rib-stiﬀened hardware pathﬁnders. The Subscale Beryllium Mirror Demonstrator (SBMD) demonstrated enabling technologies for lightweight cryogenic mirrors as a precursor to the James Webb Space Telescope (JWST). With a diameter of approximately 0.5 m and an adjustable RoC, the SBMD showed, among other requirements, a surface ﬁgure roughness of less than λ/4 p-v at λ = 633 nm under cryogenic vacuum conditions.12 Experience from SBMD was applied to the Advanced Mirror System Demonstrator (AMSD), a 1.4 m diameter rib-stiﬀened beryllium hexagonal mirror segment with four degrees of freedom (piston, tip/tilt, RoC). Cryogenic performance of this mirror also met requirements and results were used to guide JWST segment design.13 Finally, the Primary Mirror Segment Assembly (PMSA) demonstrated TRL-6 for the JWST primary mirror by verifying segment requirements in a space-equivalent thermal and vacuum environment.14, 15 Of particular relevance to the present study is the work of Budinoﬀ and Michels5 and Park et al.,16 both of which consider optimization over geometric design parameters to reduce residual ﬁgure error. Budinoﬀ and Michels demonstrate a signiﬁcant reduction in actuator-induced residual for the Spherical Primary Optical Telescope (SPOT) segment. The authors create a ﬁnite element model of the cast Pyrex mirror substrate and parameterize the back surface shape using a series of basis functions that deﬁne the substrate thickness. By coupling the parameterized model with an optimization routines, they determine the mirror substrate geometry that minimizes residual caused by the single surface-normal actuator used for RoC changes. Park et al. likewise reduce mirror surface errors via ﬁnite element modeling and geometry optimization. In this case, however, the authors attempt to mitigate errors due to gravity sag and manufacturing print-through rather than actuator eﬀects. Neither Budinoﬀ and Michels nor Park et al. consider geometry variation for active rib-stiﬀened mirrors, which is the subject of this work. 3. MIRROR MODELING APPROACH Figure 2 below shows the approach to mirror modeling. A parametric ﬁnite element mirror model lies at the center of the process. The user passes to the model a single input ﬁle that contains parameters deﬁning high-level properties of the mirror (e.g. areal density, number of actuators, substrate geometry, segment diameter, etc). These parameters are passed to model that automatically constructs a ﬁnite element representation of the desired mirror in MSC.Nastran. This work assumes the use of reaction bonded SiC as the mirror substrate, given the beneﬁts identiﬁed in Section 1. The FE solver is used to generate a number of user-selectable ﬁgures of merit; residual error is used in this work. The parametric and automated nature of the FE model allows for rapid design iteration, either by varying parameters of interest to explore the trade space or by using an optimization routine to minimize a user-deﬁned cost function. Proc. of SPIE Vol. 7731 773122-3 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms Figure 2. Design process supported by the MOST ﬁnite element mirror model. 3.1 Baseline mirror Table 1 shows the parameters of the baseline mirror used in this thesis. It consists of a 1.0 m diameter (ﬂatﬂat) substrate with 156 embedded actuators. The areal density of the SiC substrate alone is 8 kg/m2 . The embedded actuators add 2 kg/m2 , miscellaneous cabling and electronics add 1 kg/m2 , and the three kinematic bipod position actuators add 1 kg/m2 , approximately.1 The total areal density of the baseline mirror segment is therefore approximately 12 kg/m2 . Table 1. Baseline mirror parameters. Parameter Diameter (ﬂat-ﬂat) Areal density (SiC substrate only) Radius of curvature Rib cell length Rib height Rib thickness Facesheet thickness Number of actuators Actuator length Actuator length (fraction of rib cell) Baseline value 1.0 m 8 kg/m2 6m 14.4 cm 25.4 mm 1 mm 1.8 mm 156 7.2 cm 0.5 3.2 Finite element implementation Figure 3(a) shows the ﬁnite element implementation of the baseline mirror. The view is of the rib-stiﬀened back structure, with the reﬂecting surface pointing into the page. Surface-parallel electrostrictive actuators are bonded to the center of the rib cells along the edge furthest from the facesheet; they are highlighted in Figure 3(a) by dark lines. The mirror substrate is modeled as a collection of Nastran plate elements. Details of the ﬁnite element construction for a rib section are shown in Figure 3(b). The facesheet is constructed from triangular CTRIA3 elements in order to match the triangular symmetry of the facesheet regions between ribs. The ribs are modeled using quadrilateral CQUADR elements that intersect the facesheet elements at right angles. The actuators are modeled as one dimensional CBAR elements anchored to the edge nodes of the rib quadrilateral elements. Piezoelectric strains are not directly supported by MSC.Nastran, therefore a thermal analogy is used wherein the application of temperature commands and subsequent strains due to CTE eﬀects mimic the behavior of Proc. of SPIE Vol. 7731 773122-4 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms (a) (b) Figure 3. (a) Finite element mirror model for the baseline mirror and (b) rib, facesheet, and actuator detail4 showing triangular, quadrilateral, and rod elements. electrostrictive material. Ĉoté et al.17 describe the modeling process in detail and experimentally validate the thermal/piezo strain equivalence relationships. The MOST mirror model has been tested for proper convergence with increasing mesh density. Furthermore, the model has been validated by comparing key ﬁgures of merit against empirical data. Speciﬁcally, the fundamental frequency of the mirror model matched that of a laboratory article within approximately 2% and the residual wavefront error for a given RoC command matched within approximately 7%.7 3.3 Command and residual calculation This work assumes that low-order changes to the mirror shape are commanded in order to correct quasi-static disturbances such as thermal distortion, therefore the control bandwidth is relatively low. A radius of curvature change (ΔRoC) of 1 mm is chosen as a representative maneuver from which the actuator-induced residual is calculated. The embedded actuators can be used to command other low-order shape changes, however ΔRoC is used as a baseline in this case. If the mirror surface shape is parameterized by a desired set of node displacements z̄ in the optical axis direction z, then the actuator commands ū for achieving those displacements are given by the solution to, H ū + z̄ = 0̄ (1) Here H is the matrix of inﬂuence functions calculated by displacing each actuator individually using a known input and measuring the resulting mirror displacement at each surface node. Figure 4(a) shows an inﬂuence function for one actuator. Solving (1) for the commands ū+1 mm needed to actuate 1 mm ΔRoC (i.e. the node displacements z̄+1 mm ) gives, −1 T ū+1 mm = − H T H H z̄+1 mm (2) −1 T A because H is non-square; the number where it is necessary to use the Moore-Penrose pseudoinverse AT A of surface nodes is typically much greater than the number of embedded actuators. Due to the discrete nature of the embedded piezoelectric actuators, the actual mirror displacements resulting from the application of the command set (2) and the desired mirror displacements z̄+1 mm will not be equivalent (see Figure 4). The diﬀerence between the desired and actual surface displacements in the presence of a 1 mm ΔRoC command is the actuator-induced residual error, shown in Figure 4(b). The following sections address changes in actuator and substrate geometry that mitigate actuator quilting in rib-stiﬀened mirrors. Proc. of SPIE Vol. 7731 773122-5 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms Residual (1mm ΔRoC) Influence function (L = 7.2 cm) 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −20 [nm] y [m] y [m] 20 0 −40 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −60 −80 −0.5 −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −100 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 x [m] x [m] (a) (b) Figure 4. (a) Inﬂuence function, i.e. the mirror displacement due to a single 7.2 cm long actuator, and (b) residual error [nm] for the baseline mirror after a 1 mm ΔRoC 4. ACTUATOR LENGTH VARIATION When exploring variations in actuator geometry, changing actuator length is a natural place to begin. Gray4 showed that in rib-stiﬀened mirrors of the type discussed here, increasing actuator length generally reduces actuator-induced residual error. This section presents additional results using the MOST mirror model and interpret ﬁndings by considering actuator inﬂuence functions. Actuator length is an adjustable parameter in the ﬁnite element mirror model used for this analysis. A range of actuator lengths from 2.4 cm to 12.0 cm was modeled, as given in Table 2. Note that each actuator is modeled Table 2. Modeled range of actuator lengths in units of physical distance, number of rib elements, and fraction of rib cell. Note that for the baseline mirror, the rib cell length is 14.4 cm. Physical length [cm] 2.4 4.8 7.2 9.6 12.0 No. of rib elements [#] 2 4 6 8 10 Rib cell fraction [-] 0.17 0.33 0.50 0.67 0.83 as a single bar element that spans an integer number of quadrilateral elements in the mirror rib. The sub-section of a rib that lies between two rib intersections is termed here a rib cell. Therefore each rib cell houses a single actuator. In addition to deﬁning actuator length by units of physical distance, it is possible to deﬁne the actuator according to the fraction of its rib cell that it occupies. The arrangement of an actuator within a rib cell is shown for three diﬀerent lengths in Figure 5. Given the above range of actuator lengths, the actuator-induced residual error was calculated in each case for ΔRoC = 1 mm. The results are given in Figure 6. Increasing the actuator length from the baseline value of 7.2 cm to 12.0 cm decreases the actuator quilting from 25.4 nm RMS to 7.2 nm RMS—a 72% reduction. Likewise, moving from 2.4 cm actuators (quilting = 43.3 nm RMS) to 12.0 cm actuators results in a 83% reduction. The reduced residual for longer actuators can be seen directly in Figure 6(b), which shows slices through the y = 0 Proc. of SPIE Vol. 7731 773122-6 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms (a) (b) (c) Figure 5. Varying actuator length l in the MOST model: (a) l = 2.4 cm, (b) l = 7.2 cm (the baseline actuator length), and (c) l = 12.0 cm Quilting vs. Actuator Length Rib cell size = 14.4 cm Residual vs. Position cross−section at y = 0 45 200 40 150 35 100 30 50 L Residual [nm] Quilting [nm RMS] act 25 20 0 −50 15 −100 10 −150 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 = 7.2 cm (50%) Lact = 12.0 cm (83%) −200 −0.5 −0.4 Normalized actuator length [fraction of rib cell] (a) −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Radial distance [m] (b) Figure 6. (a) Actuator-induced residual error [nm RMS] as a function of actuator length and (b) slice of the residual contour for l = 7.2 cm and l = 12.0 cm at y = 0 showing decreased residual for the longer actuator case. In (b) the vertical dashed lines represent the location of rib intersections. line of the residual contour (e.g. Figure 4). Comparing the baseline mirror (l = 7.2 cm) to the best performing mirror (l = 12.0 cm), there is a clear reduction in the magnitude residual error at all points within the mirror surface. Looking at the changing inﬂuence function of a given actuator when altering its length can oﬀer additional insights into the reduction in actuator quilting. Two inﬂuence functions for the same actuator with diﬀerent lengths are shown in Figure 7. Figure 7(a) shows the inﬂuence function for l = 7.2 cm and Figure 7(b) shows the inﬂuence function for l = 12.0 cm. Comparing those two ﬁgures, the longer actuator yields a larger region of inﬂuence, as indicated by the wider contour spacing. Taking horizontal and vertical slices along the dashed lines indicated produces Figure 7(c) and (d), respectively. Considering Figure 7(c), a slice of the contour at y = 0 clearly reveals that the longer actuator has the eﬀect of broadening the region of inﬂuence along the rib in which the actuator is situated. Likewise, Figure 7(d) shows a similar, albeit less pronounced, eﬀect in the direction perpendicular to the rib (i.e. into the center of the triangular facesheet region on either side of the Proc. of SPIE Vol. 7731 773122-7 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms rib). By increasing the inﬂuence function in two dimensions, lengthening the actuator provides better actuation coverage over the mirror surface. The result is an ability to command low-order prescription changes with reduced actuator-induced quilting. Influence function (L = 12.0 cm) 0.1 0.1 0.05 0.05 y [m] y [m] Influence function (L = 7.2 cm) 0 0 −0.05 −0.05 −0.1 −0.1 0 0.05 0.1 0.15 0 0.05 x [m] (a) x 10 −9 1.8 L = 7.2 cm (50%) L = 12.0 cm (83%) 1.6 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.02 0.04 0.06 0.08 x [m] 0.1 0.12 0.14 x 10 Influence function vertical slice L = 7.2 cm (50%) L = 12.0 cm (83%) 1.6 1.4 0.2 0 0.15 (b) Influence function horizontal slice −9 1.8 0.1 x [m] 0.2 −0.1 (c) −0.05 0 y [m] 0.05 0.1 (d) Figure 7. (a) Inﬂuence functions for actuator length l = 7.2 cm and (b) l = 12.0 cm; the view is zoomed in and the thick horizontal bar represents the actuator location. (c) Comparing horizontal slices of the two inﬂuence functions and (d) comparing vertical slices of the two inﬂuence functions. 5. SUBSTRATE GEOMETRY VARIATION The above section considered the eﬀects of lengthening surface-parallel actuators in rib stiﬀened mirrors. Beneﬁts in terms of reduced actuator quilting are seen in the case of actuators that occupy a larger portion of the rib cell in which they reside. Lengthening actuators, however, may not always be feasible due to manufacturing constraints. This section presents altering substrate geometry as a means of reducing actuator-induced residual error while keeping actuator length constant. Proc. of SPIE Vol. 7731 773122-8 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms As mentioned above, Budinoﬀ and Michels5 reduced actuator-induced residual by altering the substrate geometry to spread actuator loads more evenly. A similar approach for the case of rib-stiﬀened mirrors is presented here. The “basis function” describing the substrate geometry in this case is a sinusoidal variation in rib height with a period equal to a single rib cell (see Figure 8). The rib shape is parameterized by the amplitude of the variation a, where the modeled values are a = {0, 2.5, 5, 7.5, 10, 12.5, 15, 17.5, 20} mm. In each case the mass of the substrate stays constant, therefore so does the mirror areal density. (a) (b) Figure 8. (a) Rib shape with a = 0 mm (i.e. constant rib height) and (b) rib shape with a = 10 mm (sinusoidal variation in rib height). As before, the actuator-induced residual was calculated for each a value given a baseline maneuver of 1 mm ΔRoC. The results are shown in Figure 9. The shaping amplitude value that minimizes the actuatorinduced residual is a = 12.5 mm. As a increases and material is moved from the center of the ribs toward the rib intersections, each actuator is eﬀectively lengthened, which decreases quilting for the reasons discussed in Section 4. However if too much material is removed, the ribs lose their stiﬀening qualities, which is detrimental to actuator-induced residual. For this mirror, a = 12.5 mm balances these two eﬀects. Increasing the shaping amplitude from a = 0 mm (25.4 nm RMS residual) to a = 12.5 mm (18.6 nm RMS residual) decreases actuatorinduced quilting by 27%. Quilting vs. Shaping Amplitude Sinusoidal Shaping Function Residual vs. Position cross−section at y = 0 26 200 25 150 24 100 23 50 Residual [nm] Quilting [nm RMS] a = 0 mm a = −12.5 mm 22 21 0 −50 20 −100 19 −150 18 0 2.5 5 7.5 10 12.5 15 17.5 20 −200 −0.5 −0.4 Shaping amplitude, a [mm] −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Radial distance [m] (a) (b) Figure 9. (a) Actuator-induced residual error [nm RMS] as a function of sinusoidal rib shaping amplitude (b) slice of the residual contour for a = 0 mm and a = −12.5 mm at y = 0. In (b) the vertical dashed lines represent the location of rib intersections. Proc. of SPIE Vol. 7731 773122-9 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms As before, additional insights can be gained by examining the inﬂuence functions of the baseline and optimized mirrors. These are shown in Figure 10. Comparing the contour plots of the inﬂuence functions—Figures 10(a) and (b)—the diﬀerence between the baseline and optimized case is not as pronounced as when varying actuator length. However, the larger contour spacing in Figure 10(b) indicates a broader inﬂuence function for the a = 12.5 case. This is reﬂected in the horizontal and vertical slices through the contours, shown in Figures 10(c) and (d), respectively. While the horizontal inﬂuence function expansion along ribs is smaller than the case of lengthening actuators, it nonetheless exists when shaping is added. The same applies for vertical spread of the inﬂuence function into adjacent facesheet regions. Thus while the results are less drastic than actuator lengthening, sinusoidal rib shaping does have beneﬁts in terms of inﬂuence function spread, which in turn reduces actuator-induced residual. Influence function (a = 12.5 mm) 0.1 0.1 0.05 0.05 y [m] y [m] Influence function (a = 0 mm) 0 0 −0.05 −0.05 −0.1 −0.1 0 0.05 0.1 0.15 0 x [m] −9 1.8 a = 0 mm a = 12.5 mm 1.6 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.02 0.04 0.06 0.08 y [m] 0.1 0.12 0.14 x 10 Influence function vertical slice a = 0 mm a = 12.5 mm 1.6 1.4 0.2 0 0.15 (b) Influence function horizontal slice −9 x 10 0.1 x [m] (a) 1.8 0.05 0.2 −0.1 (c) −0.05 0 y [m] 0.05 0.1 (d) Figure 10. (a) Inﬂuence functions for constant rib height a = 0 mm and (b) optimal shaping amplitude a = 12.5 mm. (c) Comparing horizontal slices of the two inﬂuence functions and (d) comparing vertical slices of the two inﬂuence functions. Proc. of SPIE Vol. 7731 773122-10 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms 6. CONCLUSIONS AND FUTURE WORK It has been demonstrated that changes to mirror substrate and actuator geometry can reduce actuator-induced residual errors in rib-stiﬀened active primary mirrors. Increasing the length of embedded surface-parallel actuator from 50% to 83% of a rib cell decreases residual errors for 1 mm ΔRoC by 73% for the mirror studied. This is due to a broadening of the localized actuator inﬂuence functions, providing better coverage over the mirror surface for low-order prescription changes. Likewise, initial variations in substrate geometry show promise for reducing actuator-induced residual. A sinusoidally-varying height pattern was added to the ribs. Compared to the baseline (constant rib height), actuator quilting was reduced 27% by using an aplitude variation 12.5 mm. While incorporating geometric variations of this type in rib-stiﬀened SiC substrates may pose manufacturing diﬃculties, it nonetheless could reduce actuator-induced residual while keeping areal density and number of actuators constant. 6.1 Patch actuators Surface-parallel actuators of the type discussed to this point take advantage of the d33 eﬀect of electrostrictive materials—i.e. that voltage across a given axis will produce a strain in that same axis. Others have investigated the use of the d31 property of electrostrictive actuators, whereby the material undergoes strain in a plane normal to the voltage application. In such arrangements, two dimensional “patch” actuators bonded to reﬂective surfaces are used to locally alter mirror curvature. Dürr et al.18 and Shepherd et al.10 provide two examples. Electrostrictive patch actuators would be a natural addition to rib-stiﬀened mirrors. Surface-parallel rod type actuators create inﬂuence functions that are situated on the ribs, which leaves pockets of deﬂection in the inter-rib locations. Placing a patch actuator in each triangular region of the face sheet between ribs would provide another set of inﬂuence functions that could aid in the actuation of smoother shape changes. The location of a single patch actuator on the facesheet opposite the reﬂecting surface is shown in Figure 11. In an actual implementation, the entire back side of the mirror facesheet would be covered with patch actuators to augment the cylindrical actuators distributed among the ribs. (a) (b) Figure 11. (a) Sample patch actuator implemented using 2D triangular plate elements and (b) detailed view of the patch actuator. Figure 12 shows the inﬂuence function of the single patch actuator depicted in Figure 11. These results were obtained by using the MOST mirror model and adding a series of 2D CTRIA3 elements to serve as the patch actuator. Following the method used for the cylindrical actuators discussed above, the patch actuator uses thermal analogy to model piezoelectric strains. The resulting inﬂuence function is highly localized within the triangular inter-rib region of the facesheet. Future work includes modeling patch actuators throughout the mirror and incorporating their inﬂuence functions into the command set (2) to determine the eﬀect on actuator-induced residual error. Such actuators could also help mitigate manufacturing-induced print-through eﬀects as well.8 Proc. of SPIE Vol. 7731 773122-11 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms Influence function (patch actuator) Influence function (patch actuator) 0.5 0.4 0.1 0.3 0.2 0.05 y [m] y [m] 0.1 0 0 −0.1 −0.05 −0.2 −0.3 −0.1 −0.4 −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 x [m] x [m] (a) (b) Figure 12. (a) Inﬂuence function of a patch actuator and (b) magniﬁed view of the inﬂuence function. 6.2 Optimized rib-stiﬀened mirror geometry Key to the work of Budinoﬀ and Michels on SPOT was the identiﬁcation of a suitable set of basis functions that were used parameterize and then optimize the shape of the mirror substrate. Selecting a set of basis functions is therefore an essential ﬁrst step in performing geometric optimization. The results presented above point to sinusoidal along-rib shaping as a possible geometric variation to use when optimizing the shape of rib-stiﬀened active mirrors. Another possibility would be to add smooth blending at the rib-facesheet junction. In current rib-stiﬀened mirrors, the rib joins the facesheet at a right angle. There is therefore little ability to spread the actuator load from the rib into the adjoining facesheet regions. An example of blended geometry is depicted in Figure 13. (a) (b) Figure 13. (a) Standard rib-facesheet intersection (right angle) and (b) blended rib-facesheet intersection. The facesheet is horizontal on the bottom, while the rib is the piece that rises vertically. Future work would conduct a SPOT-type optimization5 over mirror shape using the sinusoidal rib basis function identiﬁed here and rib-to-facesheet blending. The result would be a design evolution like the one shown in Figure 14. Figure 14(a) represents the current state of rib-stiﬀened mirror design, in which ribs are constant height and there is no rib-to-facesheet blending. Figure 14(b) shows a mirror incorporating the sinusoidal rib shaping pattern discussed here, which spreads the embedded actuator loads and decreases actuator-induced residual. Figure 14(c) depicts the end-state of an optimization eﬀort: a mirror substrate featuring both rib shaping and rib-to-facesheet blending, both of which reduce actuator-induced residual without increasing areal density. The resulting new design would present manufacturing challenges, given the current methods of casting reaction bonded SiC mirrors. Furthermore, the stiﬀening eﬀect of the ribs is likely compromised when blending is Proc. of SPIE Vol. 7731 773122-12 Downloaded from SPIE Digital Library on 11 Feb 2011 to 18.51.1.125. Terms of Use: http://spiedl.org/terms (a) (b) (c) Figure 14. (a) Standard mirror substrate with constant rib height, (b) mirror substrate with along-rib shaping, and (c) mirror substrate with along-rib shaping and rib-to-facesheet blending. In all cases the facesheet is oriented with its reﬂecting surface downward. introduced, therefore a constraint on minimum stiﬀness would need to be included in the optimization. However, the above results demonstrate that a process of judicious geometric variation could yield a new set of lightweight mirrors with reduced actuator-induced residual error. REFERENCES [1] Ealey, M. A. and Wellman, J. 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